Moving kinks and nanopterons in the nonlinear Klein–Gordon lattice

We study moving topological solitons (kinks and antikinks) in the nonlinear Klein–Gordon chain. These solitons are shown to exist with both monotonic (non-oscillating) and oscillating asymptotics (tails). Using the pseudo-spectral method, the (anti)kink solutions with oscillating background (so-called nanopterons) are found as travelling waves of permanent profile propagating with constant velocity. Each of these solutions may be considered as a bound state of an (anti)kink with a background nonlinear periodic wave, so that the wave “pushes” the (anti)kink over the Peierls–Nabarro barrier. The stability of these bound states is confirmed numerically. Travelling-wave solutions of permanent profile are shown to exist depending on the convexity of the on-site (substrate) potential. The set of velocities at which the (anti)kinks with monotonic asymptotics propagate freely is calculated. We also find moving non-oscillating (anti)kink profiles with higher topological charges, each of which appears to be the bound state of (anti)kinks with lower topological charge (∣Q∣=1).